Theorem cgraswap 25712

Description: Swap rays in a congruence relation. Theorem 11.9 of [Schwabhauser] p. 96. (Contributed by Thierry Arnoux, 5-Mar-2020.)

Hypotheses

Ref	Expression
cgraid.p	|- P = ( Base ` G )
cgraid.i	|- I = (Itv ` G )
cgraid.g	|- ( ph -> G e. TarskiG )
cgraid.k	|- K = (hlG ` G )
cgraid.a	|- ( ph -> A e. P )
cgraid.b	|- ( ph -> B e. P )
cgraid.c	|- ( ph -> C e. P )
cgraid.1	|- ( ph -> A =/= B )
cgraid.2	|- ( ph -> B =/= C )

Assertion

Ref	Expression
cgraswap	|- ( ph -> <" A B C "> (cgrA ` G ) <" C B A "> )

Proof of Theorem cgraswap

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgraid.p	. . . . . . . 8 |- P = ( Base ` G )
2		eqid 2622	. . . . . . . 8 |- ( dist ` G ) = ( dist ` G )
3		cgraid.i	. . . . . . . 8 |- I = (Itv ` G )
4		cgraid.g	. . . . . . . . 9 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> G e. TarskiG )
6		cgraid.b	. . . . . . . . 9 |- ( ph -> B e. P )
7	6	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> B e. P )
8		simpllr 799	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> x e. P )
9		cgraid.a	. . . . . . . . 9 |- ( ph -> A e. P )
10	9	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> A e. P )
11		simprlr 803	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) )
12	1, 2, 3, 5, 7, 8, 7, 10, 11	tgcgrcomlr 25375	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( x ( dist ` G ) B ) = ( A ( dist ` G ) B ) )
13	12	eqcomd 2628	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( A ( dist ` G ) B ) = ( x ( dist ` G ) B ) )
14		simprrr 805	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) )
15	14	eqcomd 2628	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( B ( dist ` G ) C ) = ( B ( dist ` G ) y ) )
16		simplr 792	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> y e. P )
17		cgraid.c	. . . . . . . . 9 |- ( ph -> C e. P )
18	17	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> C e. P )
19		eqid 2622	. . . . . . . . 9 |- (LineG ` G ) = (LineG ` G )
20		eqid 2622	. . . . . . . . 9 |- (cgrG ` G ) = (cgrG ` G )
21		cgraid.k	. . . . . . . . . . . 12 |- K = (hlG ` G )
22		simprll 802	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> x ( K ` B ) C )
23	1, 3, 21, 8, 18, 7, 5, 19, 22	hlln 25502	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> x e. ( C (LineG ` G ) B ) )
24	23	orcd 407	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( x e. ( C (LineG ` G ) B ) \/ C = B ) )
25	1, 19, 3, 5, 18, 7, 8, 24	colrot1 25454	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( C e. ( B (LineG ` G ) x ) \/ B = x ) )
26		eqid 2622	. . . . . . . . . . 11 |- (≤G ` G ) = (≤G ` G )
27	1, 3, 21, 8, 18, 7, 5	ishlg 25497	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( x ( K ` B ) C <-> ( x =/= B /\ C =/= B /\ ( x e. ( B I C ) \/ C e. ( B I x ) ) ) ) )
28	22, 27	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( x =/= B /\ C =/= B /\ ( x e. ( B I C ) \/ C e. ( B I x ) ) ) )
29	28	simp3d 1075	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( x e. ( B I C ) \/ C e. ( B I x ) ) )
30	29	orcomd 403	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( C e. ( B I x ) \/ x e. ( B I C ) ) )
31		simprrl 804	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> y ( K ` B ) A )
32	1, 3, 21, 16, 10, 7, 5	ishlg 25497	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( y ( K ` B ) A <-> ( y =/= B /\ A =/= B /\ ( y e. ( B I A ) \/ A e. ( B I y ) ) ) ) )
33	31, 32	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( y =/= B /\ A =/= B /\ ( y e. ( B I A ) \/ A e. ( B I y ) ) ) )
34	33	simp3d 1075	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( y e. ( B I A ) \/ A e. ( B I y ) ) )
35	1, 2, 3, 26, 5, 7, 18, 8, 7, 7, 16, 10, 30, 34, 15, 11	tgcgrsub2 25490	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( C ( dist ` G ) x ) = ( y ( dist ` G ) A ) )
36	1, 2, 20, 5, 7, 18, 8, 7, 16, 10, 15, 35, 12	trgcgr 25411	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> <" B C x "> (cgrG ` G ) <" B y A "> )
37	1, 2, 3, 5, 18, 16	axtgcgrrflx 25361	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( C ( dist ` G ) y ) = ( y ( dist ` G ) C ) )
38		cgraid.2	. . . . . . . . . 10 |- ( ph -> B =/= C )
39	38	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> B =/= C )
40	1, 19, 3, 5, 7, 18, 8, 20, 7, 16, 2, 16, 10, 18, 25, 36, 14, 37, 39	tgfscgr 25463	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( x ( dist ` G ) y ) = ( A ( dist ` G ) C ) )
41	1, 2, 3, 5, 8, 16, 10, 18, 40	tgcgrcomlr 25375	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( y ( dist ` G ) x ) = ( C ( dist ` G ) A ) )
42	41	eqcomd 2628	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) )
43	13, 15, 42	3jca 1242	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( ( A ( dist ` G ) B ) = ( x ( dist ` G ) B ) /\ ( B ( dist ` G ) C ) = ( B ( dist ` G ) y ) /\ ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) ) )
44	4	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ y e. P ) -> G e. TarskiG )
45	9	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ y e. P ) -> A e. P )
46	6	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ y e. P ) -> B e. P )
47	17	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. P /\ y e. P ) -> C e. P )
48		simp2 1062	. . . . . . . 8 |- ( ( ph /\ x e. P /\ y e. P ) -> x e. P )
49		simp3 1063	. . . . . . . 8 |- ( ( ph /\ x e. P /\ y e. P ) -> y e. P )
50	1, 2, 20, 44, 45, 46, 47, 48, 46, 49	trgcgrg 25410	. . . . . . 7 |- ( ( ph /\ x e. P /\ y e. P ) -> ( <" A B C "> (cgrG ` G ) <" x B y "> <-> ( ( A ( dist ` G ) B ) = ( x ( dist ` G ) B ) /\ ( B ( dist ` G ) C ) = ( B ( dist ` G ) y ) /\ ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) ) ) )
51	50	3expa 1265	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ y e. P ) -> ( <" A B C "> (cgrG ` G ) <" x B y "> <-> ( ( A ( dist ` G ) B ) = ( x ( dist ` G ) B ) /\ ( B ( dist ` G ) C ) = ( B ( dist ` G ) y ) /\ ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) ) ) )
52	51	adantr 481	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( <" A B C "> (cgrG ` G ) <" x B y "> <-> ( ( A ( dist ` G ) B ) = ( x ( dist ` G ) B ) /\ ( B ( dist ` G ) C ) = ( B ( dist ` G ) y ) /\ ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) ) ) )
53	43, 52	mpbird 247	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> <" A B C "> (cgrG ` G ) <" x B y "> )
54	53, 22, 31	3jca 1242	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( <" A B C "> (cgrG ` G ) <" x B y "> /\ x ( K ` B ) C /\ y ( K ` B ) A ) )
55	38	necomd 2849	. . . . 5 |- ( ph -> C =/= B )
56		cgraid.1	. . . . . 6 |- ( ph -> A =/= B )
57	56	necomd 2849	. . . . 5 |- ( ph -> B =/= A )
58	1, 3, 21, 6, 6, 9, 4, 17, 2, 55, 57	hlcgrex 25511	. . . 4 |- ( ph -> E. x e. P ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) )
59	1, 3, 21, 6, 6, 17, 4, 9, 2, 56, 38	hlcgrex 25511	. . . 4 |- ( ph -> E. y e. P ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) )
60		reeanv 3107	. . . 4 |- ( E. x e. P E. y e. P ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) <-> ( E. x e. P ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ E. y e. P ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) )
61	58, 59, 60	sylanbrc 698	. . 3 |- ( ph -> E. x e. P E. y e. P ( ( x ( K ` B ) C /\ ( B ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` B ) A /\ ( B ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) )
62	54, 61	reximddv2 3020	. 2 |- ( ph -> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x B y "> /\ x ( K ` B ) C /\ y ( K ` B ) A ) )
63	1, 3, 21, 4, 9, 6, 17, 17, 6, 9	iscgra 25701	. 2 |- ( ph -> ( <" A B C "> (cgrA ` G ) <" C B A "> <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x B y "> /\ x ( K ` B ) C /\ y ( K ` B ) A ) ) )
64	62, 63	mpbird 247	1 |- ( ph -> <" A B C "> (cgrA ` G ) <" C B A "> )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  ≤Gcleg 25477  hlGchlg 25495  cgrAccgra 25699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-cgra 25700

This theorem is referenced by:  cgraswaplr  25716  oacgr  25723  tgasa1  25739  isoas  25744